Monday Math Problem: The Answer to “Divisibility of Your Mom”

March 12, 2010

Every Monday, I’ll be posting a math problem on this site; every Friday, I’ll post the answer to that week’s problem. Give this one a shot and post your solution in the comments!

This Week’s Math Problem:

If m is your mother’s age rounded to the nearest integer, and there is a number n such that n is the square of m, and 45 < m < 65, what are all of the possible numbers of unique prime factors n could have?

The Answer:

1, 2, and 3

I was inspired to write this question by a number of SAT and GMAT questions that are really more reading comprehension questions than math questions. (“The Sequence S is such that, when n > 1, any term Sn is equal to S(n-1) + 1″ being a fine example of a fancy-ass way to say “Hey guys, there’s a list of consecutive numbers!”)

In this case, your mom’s age is greater than 45 and less than 65. We want to square that number, and count its unique prime factors. Of course, squaring a number doesn’t change the number of unique prime factors — for instance, the number 14 has two prime factors, 2 and 7, and therefore also two unique prime factors. If you square 14, you get 196, which, not at all coincidentally, has the prime factors 2, 2, 7, and 7. In other words, it still has two unique prime factors (it just has some duplicates). So, we can ignore the part of the mom problem about squaring.

If you wrote out a list of your mom’s possible ages, from 46 to 64, you’d quickly notice that there are many prime numbers in there. All primes have exactly one prime factor, so those are done with. As you start breaking down the other numbers, you’ll find plenty with two unique primes (46 = 23 x 2, 48 = 24 x 3, etc.), and one with three unique primes (60 = 22 x 3 x 5). Squaring these numbers won’t change the number of unique primes, so the possible answers are 1, 2, and 3.

Posted by jen | Filed Under Monday Math Problems 

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